for a fixed \(f=m/(n+m)\) . This asymptotic result eq:valim can be obtained by using the asymptotic behavior of polygamma functions [1]}
\(\psi _0(x)&=&\Theta \left(\ln (x)-\frac{1}{2 x}\right),\qquad x\rightarrow \infty \\\psi _{j}(x)&=&\Theta \left(\frac{1}{x^{j}}\right),\qquad x\rightarrow \infty ,~~~~~~j\ge 1,\) 
